The Relevance of Logarithmic Models for Population Interaction

Abstract

The central theme of this article can be given a very simple operational description, namely, that both from the point of view of developing deterministic models or simply from the point of view of looking at data, there is a considerable advantage in thinking in terms of the logarithm of population size rather than population size itself. This advantage has been frequently observed in the case of single-population models. The advantage is at least as great for more than one population. To demonstrate this, a logarithmic model is compared with classical multipopulation linear and totally linear models from two points of view. From the point of view of mathematical tractability, that is, the ease with which a model can answer questions asked of it, the logarithmic and totally linear models are equivalent and considerably more tractable than the popular linear model. The tractability of the logarithmic model is demonstrated by looking in detail at its solution for a two-population predator-prey system and by presenting a general solution for the case of any number of populations. From the point of view of ecological relevance, that is, how faithfully a model represents natural populations, the logarithmic and linear models are generally superior to the totally linear model. The logarithmic and linear models are nearly equivalent over limited ranges of population size. The model which is most relevant over less restricted ranges depends, of course, on the populations they are to represent. For a two-population predator-prey system, a comparison of the two models suggests that the logarithmic model is probably as good as, if not better than, the linear model. Some natural population data suggest the same conclusion in the case of single-population systems. In addition to comparison with other models, a number of qualitative implications of the logarithmic model are summarized. The model predicts a scalloped shape, with rounded minima and sharp peaks, for population oscillations. Under the model a single-population system cannot oscillate. Two-population systems can oscillate at one frequency, four-population systems at two frequencies, six-population systems at three frequencies, etc. The model indicates, however, that self-maintained oscillations are unlikely. It suggests that some driving force, such as added periodic variation in the relative growth rate or stochastic variation in the model parameters, is necessary for sustained oscillation.